A synchronization method of semi-markov jump neural network with measurement loss

Abstract

This invention discloses a synchronization method for a semi-Markov jumping neural network with measurement loss, belonging to the field of synchronization control technology for semi-Markov jumping neural networks. This invention can guarantee the mean-square global asymptotic stability of the error system when external disturbances are zero, and meet a preset disturbance attenuation level when external disturbances exist. The controller in this invention explicitly includes a non-fragile term structure, covering the control gain to realize error / uncertainty, avoiding the risk of "nominal design feasibility but implementation failure." This invention employs a dynamic event triggering mechanism to avoid Zeno risk at the mechanism level through "minimum trigger interval," and avoids performance degradation caused by long-term lack of updates through "maximum interval forced update." Dynamic variables achieve adaptive trigger thresholds, thereby reducing unnecessary transmissions while meeting performance requirements.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of semi-Markov jump neural network synchronization control, and particularly relates to a synchronization method of a semi-Markov jump neural network with missing measurements. BACKGROUND

[0002] In the synchronization control / estimation scenario of semi-Markov jump neural networks (Semi-Markovian jumping NNs), an engineering system is usually subject to three key constraints at the same time:

[0003] The mode residence time has a general distribution: the traditional Markov jump model often uses exponential residence time (memory-independent), and it is difficult to describe the mode evolution law of "residence time distribution non-exponential" in a real system. The semi-Markov process is explicitly used to model the jump time and residence time, and the reality of "residence time as a general distribution" is emphasized.

[0004] There are random missing measurements in the measurement signal: actual sensors / network links will have packet loss or missing measurements, so that the controller / estimator gets "incomplete measurements", which leads to deviation of the driving term of the synchronization error and misjudgment of the trigger criterion.

[0005] Network-induced transmission time delay + "Zeno risk" and "resource waste" contradiction of event-triggered communication:

[0006] If the existing event-triggered strategy does not explicitly limit the trigger interval, Zeno behavior may occur; if Zeno risk is avoided only by a conservative threshold, the communication frequency will be significantly increased, and "less transmission" and "performance index (mean square stability + attenuation)" cannot be considered.

[0007] Further, many existing methods also have two common deficiencies:

[0008] Non-fragility at the controller implementation level is not considered: control gains often have uncertainties / drifting in implementation, and if only "nominal gains" are designed, the actual system may be unstable.

[0009] The criterion is conservative: under the joint action of modes, time delays, missing measurements and network time delays, if "mode and time delay related" refinement is not performed, only very conservative stability / performance conditions can be obtained, resulting in small feasible region and poor performance index.

[0010] ​The core problem is: under the combined constraints of "semi-Markov general residence time + time-varying delay + network-induced transmission delay + missing measurement + event-triggered communication + controller gain realization uncertainty (non-fragile)," how to design a computable (LMI-solvable) control scheme that makes the master-slave system error globally asymptotically stable in the mean-square sense and satisfies the preset conditions. To improve perturbation attenuation performance while significantly reducing communication overhead and avoiding Zeno risk, a synchronization method using a semi-Markov hopping neural network with measurement loss is proposed. Summary of the Invention

[0011] The technical problem to be solved by this invention is: how to design a computable (LMI solvable) control scheme that makes the master-slave system error globally asymptotically stable in the mean square sense, while satisfying a preset disturbance attenuation level. This paper presents a synchronization method for a semi-Markov jumping neural network with measurement loss, which improves synchronization performance while significantly reducing communication overhead and avoiding Zeno risk.

[0012] like Figure 9 As shown, the present invention solves the above-mentioned technical problems through the following technical solution, and the present invention includes the following steps:

[0013] S1: Establish a main system model of a semi-Markov jump-delay complex-valued neural network with missing measurements;

[0014] S2: Construct a master-slave synchronization framework and introduce a non-fragile asynchronous controller structure to establish a slave system model;

[0015] S3: Design a dynamic event-triggered communication mechanism and construct an error system, combining the master and slave systems to construct the error system;

[0016] S4: Define asymptotic mean square synchronization and preset disturbance attenuation level Synchronization performance, and introduces lemmas as the theoretical basis;

[0017] S5: Construct a Lyapunov functional, establish a criterion for the mean-square global asymptotic stability of the error system, and a preset perturbation attenuation level. Synchronization criterion; transforming the nonlinear constraints caused by non-fragile terms into a solvable form of linear matrix inequalities, and solving for the modally dependent non-fragile asynchronous controller gain;

[0018] S6: Applying a nonfragile asynchronous controller and a dynamic event-triggered communication mechanism to an error system to achieve a nonfragile semi-Markov jump-delay complex-valued neural network. Synchronization control.

[0019] Furthermore, in step S1, the mathematical expression of the main system model is as follows:

[0020] ;

[0021] in, Indicates time The neural state vector; Given a self-feedback matrix, and for All ; and These represent the weight matrix and the time-delay connection weight matrix, respectively. It is a nonlinear neural activation function; For time-varying time delay, satisfying and ; For system output, Given a matrix, For values ​​in the interval White noise, For probability space The upper right continuous semi-Markov process.

[0022] Furthermore, the modal evolution of the semi-Markov process is characterized by the jump time, dwell time, and transition probability, as follows:

[0023] remember For the first At the next transition time, satisfying and ,make ,in Indicates from the first The next transfer time to the first Mode within the interval of the next transition time The length of stay, and Is with Independent random variables;

[0024] random sequence Construct a Markov chain with the following known transition probabilities:

[0025] ;

[0026] Correspondingly, the transition probabilities are as follows:

[0027] ;

[0028] in, and ; Indicates from time modality Transition to time modality The transfer rate, and satisfying the constraints. The corresponding transition rate matrix is ​​defined as ;

[0029] Nonlinear neural activation function The following conditions must be met and The conditions are as follows:

[0030] ;

[0031] in Given a positive scalar, ;

[0032] The initial conditions of the main system are as follows:

[0033] ;

[0034] in, , Representing the whole -Measurable, valued at The set of random variables that satisfy: .

[0035] Furthermore, in step S2, the mathematical expression of the system model is as follows:

[0036] ;

[0037] in, This represents the neural state values ​​of the system. The output vector from the system; For control input;

[0038] The initial conditions of the system are as follows:

[0039] ;

[0040] in, Meanwhile, assume random variables , and They are independent of each other.

[0041] Furthermore, the control input Using a non-fragile controller structure, the expression is as follows:

[0042] ;

[0043] ;

[0044] in, For the control gain to be designed, Indicates the first The timing of the next event trigger. This is a non-fragile term used to characterize the uncertainty in achieving the control gain. and Given a constant matrix for dimension matching, Given an unknown time-varying matrix, satisfying the constraints .

[0045] Furthermore, in step S3, the dynamic event triggering mechanism is as follows:

[0046] ;

[0047] in, This indicates the time interval between two consecutive triggering events. This represents the maximum time interval between two consecutively triggered events. It is a symmetric positive definite matrix related to the mode. For the trigger driver, a positive scalar. , It is an internal dynamic variable.

[0048] Furthermore, the internal dynamic variables The expression is as follows:

[0049] ;

[0050] For any All The derivation steps are as follows:

[0051] S31: Based on the proposed dynamic event triggering mechanism, the following inequality holds:

[0052] ;

[0053] S32: For internal dynamic variables Based on its dynamic update rules, it can be directly deduced that:

[0054] ;

[0055] S33: Calculation yields:

[0056] .

[0057] Furthermore, in step S3, the master system and the slave system are combined, and it is set that... The corresponding error system is obtained:

[0058] ;

[0059] in, This is the system output error vector.

[0060] Furthermore, in step S4, the specific processing procedure is as follows:

[0061] S41: Define asymptotic mean-square stability

[0062] When external disturbances If the error system is asymptotically stable in the mean-square sense, that is:

[0063] ;

[0064] For any initial constraint With any initial If both conditions are met, it means that the master system and the slave system have achieved asymptotic mean square synchronization;

[0065] S42: Definition synchronous

[0066] Under zero initial conditions, if for any non-zero All satisfy:

[0067] ;

[0068] This indicates that the master system and slave system are at the disturbance attenuation level. Achieved Synchronization, at this point, the error system is called Stable, and simultaneously achieving the disturbance attenuation level. ;

[0069] S43: Introducing the Lemma

[0070] Lemmas 1 through 3 are introduced as subsequent stability and The theoretical basis for the derivation of the synchronization criterion.

[0071] Furthermore, in step S5, the specific processing procedure is as follows:

[0072] S51: Based on the Lipschitz assumption of the nonlinear activation function, the Lyapunov functional of the error system is constructed as follows:

[0073] ;

[0074] Through infinitesimal generator Differentiating the Lyapunov functional, we derive a sufficient condition for the mean-square global asymptotic stability of the error system, which is the matrix inequality of Theorem 1:

[0075] ;

[0076] ;

[0077] ;

[0078] ;

[0079] in:

[0080] ;

[0081] ;

[0082] ;

[0083] ;

[0084] ;

[0085] ;

[0086] ;

[0087] ;

[0088] and, , ;

[0089] S52: Based on the conclusion of Theorem 1, we introduce... Performance metrics are obtained to achieve the error system at disturbance attenuation level ℵ. The sufficient condition for stability is the matrix inequality of Theorem 2:

[0090] ;

[0091] in, ;

[0092] S53: To address the problem in Theorem 2 where the nonlinear matrix inequalities caused by non-fragile terms cannot be directly solved, we transform them into LMI-solvable forms, resulting in the matrix inequalities of Theorem 3:

[0093] ;

[0094] in:

[0095] ;

[0096] ;

[0097] ;

[0098] The desired mode-dependent controller gain matrix is ​​obtained using the following formula:

[0099] ;

[0100] in, The positive definite matrix obtained by solving Theorem 3. The intermediate matrix is ​​obtained from Theorem 3. All matrix inequalities are solved using the standard LMI toolchain.

[0101] The present invention has the following advantages over the prior art:

[0102] 1. Synchronization performance can be guaranteed to: ensure mean-square global asymptotic stability of the error system when external disturbances are zero; and satisfy preset parameters when external disturbances exist. Disturbance attenuation level (corresponding theorem system and performance definition).

[0103] 2. Enhanced non-fragile robustness: The controller explicitly includes a non-fragile component structure to cover control gain implementation errors / uncertainties, avoiding the risk of "nominal design feasible but failure in practice".

[0104] 3. Significantly reduce communication burden and avoid Zeno risk: The dynamic event triggering mechanism avoids Zeno risk at the mechanism level through "minimum trigger interval"; avoids performance degradation caused by long-term lack of updates through "maximum interval forced update"; and the dynamic variable enables adaptive trigger threshold, thereby reducing unnecessary transmission while meeting performance requirements.

[0105] 4. Strong computability in engineering: Theorem 3 transforms the nonlinear constraints that were originally not directly solvable into LMI form and provides a gain recovery formula, so that the design can be completed using the standard LMI toolchain. Attached Figure Description

[0106] Figure 1 This is a semi-Markov process in the embodiments of the present invention. A schematic diagram of the modal evolution process;

[0107] Figure 2 This is the system state in the embodiment of the present invention. , and error trajectory diagram ( );

[0108] Figure 3 This is the system state in the embodiment of the present invention. , and error trajectory diagram ( );

[0109] Figure 4This is a random process in the embodiments of the present invention. The trajectory diagram;

[0110] Figure 5 This is the error system output in the embodiment of the present invention. The trajectory diagram;

[0111] Figure 6 This is a diagram showing the triggering time and triggering interval generated by the dynamic event triggering mechanism in this embodiment of the invention;

[0112] Figure 7 This is a diagram showing the triggering time and triggering interval generated by the static event triggering mechanism in this embodiment of the invention;

[0113] Figure 8 These are internal dynamic variables in the embodiments of this invention. The change curve;

[0114] Figure 9 This is a flowchart illustrating the synchronization method of the semi-Markov jumping neural network with measurement loss according to the present invention. Detailed Implementation

[0115] The embodiments of the present invention are described in detail below. These embodiments are implemented based on the technical solution of the present invention, and provide detailed implementation methods and specific operation processes. However, the scope of protection of the present invention is not limited to the following embodiments.

[0116] This embodiment provides a technical solution: a synchronization method for a semi-Markov jumping neural network with measurement loss, comprising the following steps:

[0117] Step S1: Establish a main system model of a semi-Markov jump-delay neural network with missing measurements.

[0118] Consider the following semi-Markov jump-delay neural network master system with missing measurements:

[0119] (1)

[0120] in, Indicates time The neural state vector; Given a self-feedback matrix, and for All ; and These represent the weight matrix and the time-delay connection weight matrix, respectively. It is a nonlinear neural activation function; For time-varying time delay, satisfying and ; For system output, Given a matrix. A stochastic process. For values ​​in the interval The white noise, satisfying the following statistical properties:

[0121] (2)

[0122] Furthermore, in probability space Above, the process It is a right-continuous semi-Markov process that takes values ​​from a known finite set. To facilitate the characterization of a semi-Markov process, let's record... For the first At the next transition time, satisfying and .make ,in Indicates from the first The next transfer time to the first Mode within the interval of the next transition time The length of stay; and Is with Independent random variables. Random sequence. Construct a Markov chain with the following known transition probabilities:

[0123] (3)

[0124] Correspondingly, its transition probability can also be expressed as:

[0125] (4)

[0126] in and ; Indicates from time modality Transition to time modality The transfer rate, and satisfying the constraints. The corresponding transition matrix is ​​defined as follows: .

[0127] Assumption 1. For the nonlinear neural activation function under consideration... ( ),satisfy:

[0128]

[0129] in, Given a known positive constant. Further settings. This setting can effectively ensure that the system under consideration (1) has an equilibrium point, thereby demonstrating the clarity and significance of the present invention.

[0130] Next, let the initial conditions of system (1) be:

[0131] (5)

[0132] in, .also, Representing the whole -Measurable, valued at The set of random variables that satisfy: .

[0133] Step S2: Construct a master-slave synchronization framework and introduce a non-fragile control structure.

[0134] The primary objective of this invention is to study resilient properties. Regarding the synchronization issue, based on the above analysis, the corresponding slave system can be established as follows:

[0135] (6)

[0136] in, This represents the neural state values ​​from system (6). The output vector from the system; For control input, its form is:

[0137] (7)

[0138] in, The control gain to be designed; it should be noted that, The definition will be given later, and Indicates the first The timing of the secondary trigger event. Additionally, non-fragile items... The structure is unknown and is as follows:

[0139] (8)

[0140] in, and Given a constant matrix for dimension matching, Given an unknown time-varying matrix, satisfying the constraints .

[0141] Furthermore, the initial conditions corresponding to system (6) are defined as follows:

[0142] (9)

[0143] in, Meanwhile, assume random variables , and They are independent of each other.

[0144] It should be noted that during the controller design process, the control gain depends on both the control and modal variables. Therefore, the control gain exhibits an asynchronous characteristic. Specifically, the control input... From nominal gain components With the additional time-varying term Composition. It is worth noting that the introduction... This is to characterize situations where the transfer rate is uncertain or the information is incomplete. This controller design can address situations where... The triggered asynchronous switching behavior is adaptively adjusted.

[0145] Step S3: Design a dynamic event-triggered communication mechanism and construct an error system.

[0146] It should be emphasized that with the continuous increase in transmitted data volume and the inherent scarcity of communication resources, efficient management and optimized allocation of communication resources are of great significance. To address this key challenge, the following dynamic event triggering mechanism is proposed:

[0147] (10)

[0148] in, It represents the time interval between two consecutive triggering events, specifically the minimum interval between the completion of the previous trigger and the occurrence of the next triggering event. This indicates the maximum time interval between two consecutively triggered events; A symmetric positive definite matrix related to the mode; positive scalar This is a scalar driven by a trigger. It's important to emphasize that when the proposed trigger condition is met, the sampled state information is updated promptly, and the controller is also updated synchronously at that moment; if the preset condition is never met, then at the constraint time... Update status information upon arrival. From this perspective, the designed event triggering scheme better meets actual engineering needs. (Alternative definition) .also, For internal dynamic variables, its expression is:

[0149] (11)

[0150] It should be noted that the dynamic variables considered are... It is non-negative, that is, for any All The derivation details are given in Algorithm 1:

[0151] Algorithm 1: Dynamic Variables The specific derivation steps are as follows:

[0152] Step 1: Based on the proposed triggering mechanism (10), the following inequality holds:

[0153]

[0154] Step 2: For dynamic variables From its dynamic update rule (11), we can directly obtain:

[0155]

[0156] Step 3: Direct calculation yields:

[0157]

[0158] Combine the master system (1) with the slave system (6), and let The corresponding error system can be obtained:

[0159] (12)

[0160] Step S4: Provide synchronization and Performance is defined and key lemmas are introduced.

[0161] Definition 1: When If the error system (12) is asymptotically stable in the mean square sense, that is:

[0162]

[0163] For any initial constraint With any initial If both conditions are met, then the master system (1) and the slave system (6) are said to have achieved asymptotic mean square synchronization.

[0164] Definition 2: Under zero initial conditions, if for any non-zero All satisfy:

[0165]

[0166] The master and slave systems (1) and (6) are then referred to by the disturbance attenuation level. Achieved Synchronization. At this point, the error system (12) is called... Stable, and simultaneously achieving the disturbance attenuation level. .

[0167] Lemma 1: Given any integrable vector (And all integrals involved are already defined) and positive definite matrices Then we have:

[0168]

[0169] in, Let be any positive integer.

[0170] Lemma 2: Let For any vector, For any matrix, It is a symmetric matrix, and To meet Positive numbers; if a matrix exists Make:

[0171]

[0172] Then the inequality holds:

[0173]

[0174] Lemma 3: Let It is a positive definite symmetric matrix; for any vector With constant If a real matrix exists Make:

[0175]

[0176] Then we can deduce:

[0177]

[0178] Step S5: Establish stability and Synchronization criteria are proposed, and controller design methods are given.

[0179] This invention aims to design a suitable error system (12). Controller. Specifically, the core objective is to construct a modal controller gain matrix of appropriate form. This allows the following two criteria to be satisfied simultaneously:

[0180] (1) When external disturbances are input At that time, the error system (12) is guaranteed to be globally asymptotically stable in the mean square sense;

[0181] (2) For any non-zero disturbance signal Under zero initial conditions, the error system (12) achieves Stability, and achieving the preset disturbance attenuation level. Mathematically, this requirement is equivalent to satisfying the following inequality:

[0182]

[0183] In this sense, the error system (12) is said to have Stability, and correspondingly achieve disturbance attenuation levels. .

[0184] Theorem 1: Under the condition that Assumption 1 holds, for the proposed controller gain matrix... If a positive definite matrix exists , , , A positive definite diagonal matrix any diagonal matrix real matrix , , as well as And there exists a positive scalar. The following constraints apply to all If the agreement is true, then when At that time, the error system (12) is globally asymptotically stable in the mean square sense.

[0185] (13)

[0186] (14)

[0187] (15)

[0188] and,

[0189] (16)

[0190] in:

[0191]

[0192]

[0193]

[0194]

[0195]

[0196]

[0197]

[0198]

[0199] and , .

[0200] prove:

[0201] The following are Lyapunov functional candidates for constructing the error system (12):

[0202] (17)

[0203] in:

[0204]

[0205]

[0206]

[0207]

[0208]

[0209] here, It is a positive definite matrix. For points... Define the infinitesimal generator as Then we have:

[0210]

[0211] Using conditional expectation and the law of total probability, we can further obtain:

[0212]

[0213]

[0214]

[0215] (18)

[0216] Furthermore, based on Taylor expansion, for any sufficiently small positive number... ,have:

[0217] (19)

[0218] From (18) and (19), we can deduce that:

[0219] (20)

[0220] And further, we obtained:

[0221] (21a)

[0222] in:

[0223]

[0224] right Differentiating the remaining components, we get:

[0225] (21b)

[0226] (21c)

[0227] (21d)

[0228] (21e)

[0229] Furthermore, for any real matrix From Lemma 1–2 and inequality (13), we can obtain:

[0230]

[0231] (twenty two)

[0232] Furthermore, by combining Lemma 3 with inequalities (14)–(15), the subsequent estimation relations can be derived:

[0233] (twenty three)

[0234] in:

[0235]

[0236] Theorem 1 also yields:

[0237] (twenty four)

[0238] Similarly, we can also obtain:

[0239] (25)

[0240] Subsequently, based on assumption 1, for any given diagonal matrix... ,have:

[0241] (26a)

[0242] (26b)

[0243] Therefore, there are two positive scalars. Therefore, we can deduce that:

[0244] (27a)

[0245] (27b)

[0246] Furthermore, for any symmetric matrix , always:

[0247] (28)

[0248] Combining (18)–(28), we get:

[0249] (29)

[0250] in:

[0251]

[0252] and As defined in (16). From the conditions in (16), we know that:

[0253]

[0254] This means when When , the error system (12) is globally asymptotically stable in the mean-square sense. In other words, when At that time, systems (1) and (6) achieve global asymptotic mean square synchronization. Q.E.D.

[0255] Theorem 2: Based on Assumption 1, for the proposed controller gain matrix... and the given positive constants If a symmetric matrix exists , , , A positive definite diagonal matrix any diagonal matrix real matrix and and positive numbers This makes the matrix inequalities (13)–(15) in Theorem 1 hold, and for all If the following matrix inequality is also satisfied, then the error system (12) can achieve a disturbance attenuation level. achieve stability:

[0256] (30)

[0257] in:

[0258]

[0259] The remaining symbols are consistent with the definitions in Theorem 1.

[0260] prove:

[0261] When inequality (30) holds, inequality (29) in Theorem 1 is also satisfied, thus showing that in Under these conditions, the error system (12) is globally asymptotically mean-square stable. For analysis... Performance, further defined:

[0262] (31)

[0263] We choose the same Lyapunov–Krasovskii functional (17). Under zero initial conditions, we have The solution pair along the error system (12) Differentiation yields:

[0264] (32)

[0265] It can be directly deduced from (32) This confirms that the error system (12) meets the corresponding disturbance attenuation level. preset Performance requirements. Q.E.D.

[0266] It also points out that: Although Theorem 2 gives the error system (12) Sufficient conditions for the synchronization problem, but due to For the unknown and Since the term is time-varying, matrix inequality (30) becomes nonlinear and cannot be solved directly using the standard MATLAB toolbox. To address this, Theorem 3 provides an effective strategy to transform the nonlinear term into a linear term.

[0267] Theorem 3: When Assumption 1 holds and zero initial conditions are satisfied, for non-zero external disturbances... If a positive definite matrix exists , , , Appropriate dimensionality diagonal matrix any diagonal matrix real matrix and positive numbers This makes constraints (13)–(15) in Theorem 1 hold, and for all If the following inequality is also satisfied, then the derived error system (12) can be at a disturbance attenuation level. accomplish stability:

[0268] (33)

[0269] in:

[0270]

[0271]

[0272]

[0273] The remaining notation is consistent with Theorem 2. Furthermore, the desired mode-dependent controller gain can be derived from:

[0274] (34)

[0275] get.

[0276] Proof: For ease of discussion, let:

[0277]

[0278] and define Based on the detailed derivation of Theorems 1–2 and the gain matrix form constructed in (33), we can directly obtain: the inequalities (13)–(15) in Theorem 1 and the condition This means that the error system (12) under study can achieve perturbation attenuation performance index. accomplish stability.

[0279] Furthermore, for any positive constant The following inequalities hold:

[0280] (35)

[0281] Furthermore, using Schur's complement lemma and inequality (33), we can deduce:

[0282] (36)

[0283] in:

[0284]

[0285]

[0286]

[0287] From this, we can obtain This holds true, and Theorem 3 is proved.

[0288] Step S6: Verify the effectiveness of the proposed method using numerical simulation.

[0289] This embodiment provides a numerical simulation example to illustrate the effectiveness of the proposed theoretical criterion. Consider a Markov network with two modes, namely... Furthermore, the system matrix can be taken as follows: , And let the correlation coefficient matrix be:

[0290]

[0291]

[0292] The time-varying time delay used Defined as Its corresponding and The transmission probability corresponding to the measurement is specified as... The nonlinear activation function used here for:

[0293]

[0294] Based on the above analysis, we can conclude that Furthermore, semi-Markov processes One form of modal evolution is as follows Figure 1 As shown. For the non-fragile items in the designed controller (7), To satisfy constraints The unknown matrix variable is specified, and the remaining parameters are as follows:

[0295]

[0296] This invention introduces a semi-Markov process. Its initial mode is set to Its trajectory is as follows: Figure 1 As shown, the corresponding transfer rate matrix is:

[0297]

[0298] Furthermore, for the delay event triggering mechanism (10), the relevant parameters can be set to... Sampling period The disturbance performance index is taken as... and set , , , , From inequalities (13) and (14) in Theorem 3, we know that there exists a feasible solution.

[0299] This embodiment provides Numerical matrix:

[0300] , ,

[0301] , ,

[0302] , ,

[0303] ,

[0304] , ,

[0305] ,

[0306] ,

[0307] Furthermore, based on the design steps given in (34), the required gain matrix can be obtained as follows:

[0308]

[0309] Therefore, according to Theorem 3, the error system (6) can meet the disturbance attenuation index. The following implementation is set stability.

[0310] In the numerical simulation, the initial conditions applied to the master system (1) and the slave system (6) are respectively selected as follows: when hour, and The time-domain trajectories of the master-slave system and the corresponding error system are as follows: Figure 2 and Figure 3 As shown. Under the above initial settings, the verification error system (12) can achieve global asymptotic mean square stability.

[0311] To study the effects of external disturbances, in Introducing disturbance signals It follows a uniform distribution in the interval [0, 0.2], and in Remove it. Simultaneously, send the signal. It is set to a standard normal distribution truncated to the interval [0, 1]. The time-varying trajectory is as follows Figure 4 As shown. Under zero initial conditions, the measurement output trajectory of the error system (12) is as follows. Figure 5As shown, the results clearly reflect the impact of external disturbances on the stability performance of the error system.

[0312] also, Figure 6 and Figure 7 The comparison results of different triggering mechanisms are given: Figure 6 For the dynamic event triggering strategy proposed in equation (11), a total of 24 triggering events were recorded; Figure 7 For the corresponding static event triggering strategy, a total of 65 triggers occurred. Table 1 further lists the results for the time-triggered strategy, which triggered a total of 101 times. Figure 8 The internal dynamic variables corresponding to the event triggering mechanism are given. The curve showing the change.

[0313] Table 1 Effective number of triggers under different triggering strategies

[0314]

[0315] It should be noted that the relevant literature is "X. Gao, Q. Wang, B. Fu, and X. Zhang. Further results on synchronization of chaotic Lur'e systems based on aperiodic time-triggered intermittent control. Communications in Nonlinear Science and Numerical Simulation, 129: 107694, 2024."

[0316] Finally, the impact of parameter variations on system performance will be further examined. Specifically, the control variable method will be used to obtain relevant conclusions:

[0317] When the minimum interval is triggered and maximum interval When changes occur, Performance indicators It fluctuated around 4.621. Therefore, it can be concluded that... It has stabilized at around 4.621, indicating that... The performance is largely unaffected by the adjustment of the trigger interval boundaries, providing a valuable reference for the parameter tuning of the control system.

[0318] Table 2 gives the parameters in Theorem 3. Different values ​​correspond to Performance metrics, the results show that when parameters When the value increases from 1 to 1.4, The performance indicators show a continuous monotonically decreasing trend, indicating that the parameters and The performance indicators show a negative correlation.

[0319] Table 2 Different parameters Values Trend of change

[0320]

[0321] With all other system parameters remaining unchanged, when the parameters in the trigger mechanism Less than the lower bound =0.99999 or exceeds the upper bound When the inequality constraints (13)–(15) and (33) have no feasible solutions, the parameters are determined accordingly. The feasible range is [0.999, 54.97]. The detailed analysis results above clearly show that selecting appropriate parameter values ​​within this feasible range is crucial for optimizing system performance.

[0322] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A synchronization method for a semi-Markov jumping neural network with measurement loss, characterized in that, Includes the following steps: S1: Establish a main system model of a semi-Markov jump-delay complex-valued neural network with missing measurements; S2: Construct a master-slave synchronization framework and introduce a non-fragile asynchronous controller structure to establish a slave system model; S3: Design a dynamic event-triggered communication mechanism and construct an error system by combining master and slave systems; S4: Define asymptotic mean square synchronization and preset perturbation attenuation level Synchronization performance, and introduces lemmas as the theoretical basis; S5: Construct a Lyapunov functional, establish a criterion for the mean-square global asymptotic stability of the error system, and a preset perturbation attenuation level. Synchronization criterion: The nonlinear constraints caused by non-fragile terms are transformed into a solvable form of linear matrix inequalities, and the modality-dependent non-fragile asynchronous controller gain is obtained by solving the problem. S6: Applying a nonfragile asynchronous controller and a dynamic event-triggered communication mechanism to an error system to achieve a nonfragile semi-Markov jump-delay complex-valued neural network. Synchronization control.

2. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 1, characterized in that, In step S1, the mathematical expression of the main system model is as follows: ； in, Indicates time The neural state vector; Given a self-feedback matrix, and for All ; and These represent the weight matrix and the time-delay connection weight matrix, respectively. It is a nonlinear neural activation function; For time-varying time delay, satisfying and ; For system output, Given a matrix, For values ​​in the interval White noise, For probability space The upper right continuous semi-Markov process.

3. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 2, characterized in that, The modal evolution of the semi-Markov process is characterized by the jump time, dwell time, and transition probability, as follows: remember For the first At the next transition time, satisfying and ,make ,in Indicates from the first The next transfer time to the first Mode within the interval of the next transition time The length of stay, and Is with Independent random variables; random sequence Construct a Markov chain with the following known transition probabilities: ； Correspondingly, the transition probabilities are as follows: ； in, and ; Indicates from time modality Transition to time modality The transfer rate, and satisfying the constraints. The corresponding transition rate matrix is ​​defined as ; Nonlinear neural activation function The following conditions must be met and The conditions are as follows: ； in Given a positive scalar, ; The initial conditions of the main system are as follows: ； in, , Representing the whole -Measurable, valued at The set of random variables that satisfy: .

4. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 3, characterized in that, In step S2, the mathematical expression of the system model is as follows: ； in, This represents the neural state values ​​of the system. The output vector from the system; For control input; The initial conditions of the system are as follows: ； in, Meanwhile, assume random variables , and They are independent of each other.

5. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 4, characterized in that, The control input Using a non-fragile controller structure, the expression is as follows: ； ； in, For the control gain to be designed, Indicates the first The timing of the next event trigger. This is a non-fragile term used to characterize the uncertainty in achieving the control gain. and Given a constant matrix for dimension matching, Given an unknown time-varying matrix, satisfying the constraints .

6. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 5, characterized in that, In step S3, the dynamic event triggering mechanism is as follows: ； in, This indicates the time interval between two consecutive triggering events. This represents the maximum time interval between two consecutively triggered events. It is a symmetric positive definite matrix related to the mode. For the trigger driver, a positive scalar. , It is an internal dynamic variable.

7. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 6, characterized in that, The internal dynamic variables The expression is as follows: ； For any All The derivation steps are as follows: S31: Based on the proposed dynamic event triggering mechanism, the following inequality holds: ； S32: For internal dynamic variables Based on its dynamic update rules, it can be directly deduced that: ； S33: Calculation yields: 。 8. The synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 7, characterized in that, In step S3, the master system and the slave system are combined, and... The corresponding error system is obtained: ； in, This is the system output error vector.

9. A synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 8, characterized in that, In step S4, the specific processing procedure is as follows: S41: Define asymptotic mean-square stability When external disturbances If the error system is asymptotically stable in the mean-square sense, that is: ； For any initial constraint With any initial If both conditions are met, it means that the master system and the slave system have achieved asymptotic mean square synchronization; S42: Definition synchronous Under zero initial conditions, if for any non-zero All satisfy: ； This indicates that the master system and slave system are at the disturbance attenuation level. Achieved Synchronization, at this point, the error system is called Stable, and simultaneously achieving the disturbance attenuation level. ; S43: Introducing the Lemma Lemmas 1 through 3 are introduced as subsequent stability and The theoretical basis for the derivation of the synchronization criterion.

10. A synchronization method for a semi-Markov jumping neural network with measurement loss according to claim 9, characterized in that, In step S5, the specific processing procedure is as follows: S51: Based on the Lipschitz assumption of the nonlinear activation function, the Lyapunov functional of the error system is constructed as follows: ； Through infinitesimal generator Differentiating the Lyapunov functional, we derive a sufficient condition for the mean-square global asymptotic stability of the error system, which is the matrix inequality of Theorem 1: ； ； ； ； in: ； ； ； ； ； ； ； ； and, , ; S52: Based on the conclusion of Theorem 1, we introduce... Performance metrics are obtained to achieve the error system at disturbance attenuation level ℵ. The sufficient condition for stability is the matrix inequality of Theorem 2: ； in, ; S53: To address the problem in Theorem 2 where the nonlinear matrix inequalities caused by non-fragile terms cannot be directly solved, we transform them into LMI-solvable forms, resulting in the matrix inequalities of Theorem 3: ； in: ； ； ； The desired mode-dependent controller gain matrix is ​​obtained using the following formula: ； in, The positive definite matrix obtained by solving Theorem 3. The intermediate matrix obtained from Theorem 3 is used to solve all matrix inequalities using the standard LMI toolchain.

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